Birch–Swinnerton–Dyer via an Operator–Fredholm Method
Birch–Swinnerton–Dyer via an Operator–Fredholm Method
We develop a clean operator-Fredholm approach to the Birch–Swinnerton–Dyer problem for an elliptic curve E/Q. On the E-isotypic Neron–Tate subspace S subset J0(N)(Q) otimes R we construct a windowed Hecke operator U_S^infty(s) whose principal parts at s=1 match those of log L(E,s). A finite R x R renormalization (R = 1 + omega_mult(N)) cancels archimedean/bad/window constants while preserving the principal part on S, hence C(p;E) = 0. Via the Hecke->Kummer bridge we obtain global Kummer classes with controlled local images; by Tate local duality the global pairing matrix reduces to the Neron–Tate Gram on S. Consequently rank E(Q) = ord_{s=1} L(E,s); uniformly in n, dim_Fp Sel_{p^n}(E/Q) <= r + Delta_loc(p;E). Finally we assemble the leading coefficient at s=1 in the standard normalizations (Omega_E, product c_p, |E(Q)_tors|) with an isogeny-invariant factor kappa. We do not assert a global identity det = L; on S_perp we use only regularized Fredholm determinants, and a spectral gap at s=1 ensures holomorphic nonvanishing.MSC 2020: Primary 11G05; Secondary 11G40, 14H52, 14G10, 47A53.Code availability: A compact demo pack (Sage/Python scripts with minimal data)is hosted on GitHub.
Birch–Swinnerton–Dyer conjecture, Néron–Tate height, compressed Fredholm determinant, Hecke operators, regulator, operator–Fredholm factorization, order of vanishing; spectral gap, L-function at s = 1, spectral gap, elliptic curves, elliptic curves;, order of vanishing
Birch–Swinnerton–Dyer conjecture, Néron–Tate height, compressed Fredholm determinant, Hecke operators, regulator, operator–Fredholm factorization, order of vanishing; spectral gap, L-function at s = 1, spectral gap, elliptic curves, elliptic curves;, order of vanishing
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